Analytic Function

A function  f(z) which is defined in D and single valued differentiable for all points of D is said to be analytic function of z in domain D. 

 Necessary condition a complex  function  to be analytic   and  Cauchy- Riemann equations.

A function f(z)= u(x,y) + i v(x,y) is said to be analytic in domain D if it satisfies the followingCauchy- Riemann equations ( For Proof see references [1][2][3])
∂u/∂x= ∂v/∂y ———–(1)
∂u/∂y =  – ∂v/∂x ——————– (2)

Example of analytic function.

                 f(z) =z2
Here, z= x + iy Then z2 = x 2 – y 2 + 2i xy
Then u(x,y) = x 2 – y 2 
and v(x,y)= 2xy
 Then  ∂u/∂x= 2x = ∂v/∂y ———–(1)
and 
∂u/∂y = -2y = – ∂v/∂x —————–(2)
Here f(z) satisfies Cauchy- Riemann equations hence function is analytic.
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References
[1] H.M. Atassi,  Analytic Functions of a Complex Variable, University of Notre Dame 
Department of Aerospace and Mechanical Engineering,
Link->https://www3.nd.edu/~atassi/Teaching/ame60612/Notes/analytic_functions.pdf
[2] Analytic Functions, Chapter 5, Link-> http://www.nhn.ou.edu/~milton/p5013/chap5.pdf
[3] Analytic Functions Link->https://application.wiley-vch.de/books/sample/3527406379_c01.pdf.


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